#### Abstract

We provide an iterative process which converges strongly to a common fixed point of finite family of asymptotically -strict pseudocontractive mappings in Banach spaces. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear operators.

#### 1. Introduction

Let be a real normed linear space with dual . A * gauge* function is a continuous and strictly increasing function satisfying and , as . The generalized duality mapping from to associated with the gauge function (see, e.g., [1]) is defined by
where denotes the duality pairing. In the case that , the duality mapping is called the * normalized duality mapping*.

Following Browder [2], we say that a Banach space has * a weakly continuous duality mapping* if there exists a gauge for which the duality mapping is single valued and weak-to-weak* sequentially continuous (i.e., if is a sequence in weakly convergent to a point , then the sequence converges weak* to ). It is known that has a weakly continuous duality mapping with a gauge function , for all .

Let be a nonempty subset of . A mapping is called * asymptotically **-strict pseudocontractive*, with sequence , (see, e.g., [3–6]) if for all , there exist and a constant such that
for all .

If denotes the identity operator, then (2) can be equivalently written as for all .

If , a real Hilbert space, it is shown by Osilike et al. [4] that (2) (and hence (3)) is equivalent to the inequality
where . is called * uniformly Lipschitz* if there exists such that for all . It is shown in [4] that an asymptotically -strict pseudocontractive mapping is uniformly Lipschitz.

The class of asymptotically -strict pseudocontractive mappings was first introduced in Hilbert spaces by Liu [5]. He proved the following theorem.

Theorem Q (see [5]). *Let be a closed convex and bounded subset of a Hilbert space . Let be completely continuous asymptotically -strict pseudocontractive mapping for some with sequence such that and . Let be a sequence generated by the modified Mann's iteration method:
**
where is a real sequence satisfying for all and some . Then, converges strongly to a fixed point of .*

The iteration scheme (5) is called * modified Mann’s iterative processes* which was introduced by Schu [7, 8] and has been used by several authors (see, e.g., [3–5, 9–17]). We observe that Liu [5] proved * strong convergence * of scheme (5) to a fixed point of asymptotically -strict pseudocontractive mapping with additional assumption that is *completely continuous*, where is said to be completely continuous if for every bounded sequence , there exists a subsequence, say of such that the sequence converges strongly to some element of the range of .

In [12], Kim and Xu studied weak convergence theorem for the class of asymptotically -strict pseudocontractive mappings in the frame work of Hilbert spaces. In fact, they proved the following.

Theorem KX (see [12]). * Let be a closed and convex subset of a Hilbert space . Let be an asymptotically -strict pseudocontractive mapping for some with sequence such that and . Let be a sequence generated by the modified Mann's iteration method:
**
where is a real sequence satisfying , for all , and . Then, converges weakly to a fixed point of .*

In 2007, Osilike et al. [13] extended Theorem KX by proving * weak convergence* of scheme (6) to a fixed point of in the frame work of uniformly smooth Banach spaces which are also uniformly convex under suitable control conditions.

In 2011, Zhang and Xie [17] extended Theorem of Osilike et al. [13] to a more general real uniformly convex Banach space with Fréchet differentiable norm. In addition, they proved * strong convergence * of scheme (5) to a fixed point of asymptotically -strict pseudocontractive mapping provided that , where .

However, we observe that the convergence obtained above is either * weak* or requiring * additional assumption * like or is completely continuous. But the requirement that is not easy to verify, as is in general unknown, and there is also an example of asymptotically -strict pseudocontractive mapping which is not completely continuous as shown below.

An example of asymptotically -strict pseudocontractive mapping which is not completely continuous.

*Example 1. * Let and . Define by , where is a real sequence satisfying , , and . Then it is shown in [13] that is asymptotically -strict pseudocontractive mapping.

Now, we show that is not completely continuous. Let be a sequence in defined by , , . Then and is given by , , . Hence, since , as , there is no subsequence of such that converges strongly to a point in , as , as . Therefore, is not completely continuous.

*Thus, one question is raised naturally:* can we obtain a scheme that converges strongly to a fixed point of asymptotically -strict pseudocontractive mappings without those additional assumptions?

It is our purpose in this paper to provide an iterative scheme which converges strongly to a common fixed point of finite family of asymptotically -strict pseudocontractive mappings in Banach spaces. The assumption that or is completely continuous is not required.

#### 2. Preliminaries

We need the following definitions from [18]. The Banach space is said to be * uniformly convex* if, given , there exists , such that, for all with and , . It is well known that , , and Sobolev spaces , (, are uniformly convex.

A Banach space is said to have a * Fréchet differentiable norm* if for all
exists and is attained uniformly in . It is well known that uniformly smooth Banach spaces has a Fréchet differentiable norm.

In order to prove our results, we need the following lemmas.

Lemma 2 (see [19]). *Let be a nonempty close convex subset of a real Banach space which has the Fréchet differentiable norm. For , let be defined for by
**
Then, and
*

It is shown in [19] that if , a real Hilbert space, then , for . In our general setting, throughout this paper we assume that .

Lemma 3. *Let be a real Banach space. Then the following inequality holds:
*

Lemma 4 (see [20]). *Let be a uniformly convex Banach space and a closed ball of . Then, there exists a continuous strictly increasing convex function with such that
**
for each and for , with .*

Lemma 5 (see [21]). *Let be a sequence of nonnegative real numbers satisfying the following relation:
**
where and satisfying the following conditions: , and . Then, .*

Lemma 6 (see [17]). *Let be a nonempty closed convex subset of a real uniformly convex Banach space which has the Fréchet differentiable norm. Let be an asymptotically -strict pseudocontractive mapping with fixed point of , . Then is demiclosed at zero, that is, if and , as , then .*

Lemma 7 (see [22]). *Let be sequences of real numbers such that there exists a subsequence of such that for all . Then there exists a nondecreasing sequence such that and the following properties are satisfied by all (sufficiently large) numbers :
**
In fact, . *

#### 3. Main Results

We now prove our main theorem.

Theorem 8. *Let be a nonempty, closed, and convex subset of a real uniformly convex Banach space which has Fréchet differentiable norm possessing a weakly sequentially continuous duality mapping from into . Let be asymptotically -strict pseudocontractive mappings for with sequences , for . Assume that is nonempty. Let be a sequence defined by and
**
where , such that , for each , , satisfying , , , , for and (, , and constants), for , Then the sequence generated by (14) converges strongly to a common fixed point of .*

*Proof. *Fix . Let and . Then, using Lemma 2 and (3) we have that

On the other hand using Lemma 4 we get that

Now substituting (16) into (15) we obtain that
since for each . Then now, from (14) and (18) we get that
where is a positive integer such that , for all , for some . Therefore, by induction,
which implies that and hence are bounded.

Furthermore, from (14), Lemma 3, and (17) we get that
for some .

Now, the rest of the proof is divided into two parts.*Case **1.* Suppose that there exists such that is decreasing for all . Then we have that is convergent. Then from (21) and the assumptions on , , and we have that , as , which implies that
for . Then from (14) we obtain that
Again, from (23) we get that
as . Thus, (24) and (25) imply that
Therefore, since each , for , is uniformly -Lipschitzian and
we have from (23), (26), and uniform continuity of that
for each . Furthermore, the fact that is bounded and is reflexive implies that we can choose a subsequence of such that and
Now, from (26) we get that and from Lemma 6 we have that , for each . Hence, . Therefore, putting in (29) and using the fact that is weakly sequentially continuous we immediately obtain that . Again, putting in inequality (22), we get that
and, hence, it follows from (30) and Lemma 5 that , as . Consequently, .*Case **2.* Suppose that there exists a subsequence of such that
for all . Then, by Lemma 7, there exists a nondecreasing sequence such that , and for all . Then from (21) and following the method of Case 1, we get that
for each . Thus, again following the method of Case 1, we obtain that and , as , for each and there exists such that
Then now, putting in (22) we have that
Since , (34) implies that
Moreover, since , inequality (35) gives that
Then, from (33) and the fact that , we obtain that , as . This together with (34) gives that , as . But , for all ; thus we obtain that . Therefore, from the above two cases, we can conclude that converges strongly to an element of and the proof is complete.

If, in Theorem 8, we assume a single asymptotically -strict pseudocontractive mapping we get the following corollary.

Corollary 9. *Let be a nonempty, closed, and convex subset of a real uniformly convex Banach space which has Fréchet differentiable norm possessing a weakly sequentially continuous duality mapping from into . Let be an asymptotically -strict pseudocontractive mapping for with sequences . Assume that is nonempty. Let be a sequence defined by and
**
where , satisfying , , , and (, , and constants). Then the sequence generated by (37) converges strongly to a fixed point of .*

*Proof. *Putting in (14), we get that and the scheme reduces to scheme (37) and following the method of proof of Theorem 8 we get that (see (21) and (22))
for some . Now, considering cases, as in the proof of Theorem 8, we obtain the required result.

Corollary 10. *Let be a nonempty, closed, and convex subset of , . Let be asymptotically -strict pseudocontractive mappings for with sequences , for . Assume that is nonempty. Let be a sequence defined by and
**
where , such that , for each , , satisfying , , , and (for , , and constants), for . Then the sequence generated by (39) converges strongly to a common fixed point of .*

*Proof. *We note that , , spaces are uniformly convex which have Fréchet differentiable norm possessing a weakly sequentially continuous duality mapping from into (see, e.g., [18]). Thus, the result follows from Theorem 8.

Corollary 11. *Let be a nonempty, closed, and convex subset of , . Let be an asymptotically -strict pseudocontractive mapping for some with sequences . Assume that is nonempty. Let be a sequence defined by and
**
where , and (for , , and constants) satisfying , and . Then the sequence converges strongly to a fixed point of . *

If in Theorem 8 we have that , a real Hilbert space, then is uniformly convex with Fréchet differentiable norm possessing a weakly sequentially continuous duality mapping. Thus, we have the following corollary.

Corollary 12. *Let be a nonempty, closed, and convex subset of a real Hilbert space . Let be asymptotically -strict pseudocontractive mappings for with sequences , for . Assume that is nonempty. Let be a sequence defined by and
**
where , such that , for each , , satisfying , , , and (for , , and constants), for . Then the sequence generated by (41) converges strongly to a common fixed point of .*

Corollary 13. *Let be a nonempty, closed, and convex subset of a real Hilbert space . Let be an asymptotically -strict pseudocontractive mapping for some with sequences . Assume that is nonempty. Let be a sequence defined by and
**
where , and (for , , and constants) satisfying , and . Then the sequence converges strongly to a fixed point of .*

*Remark 14. *We note that Corollary 9 generalizes several recent results of this nature. Particularly, it extends Theorem KX of [12], Theorem 2 of Liu [5], and corresponding theorem of Schu [7] in the sense that our convergence is *strong* in more general Banach spaces possessing weakly sequentially continuous duality mappings without the requirement that be completely continuous.

*Remark 15. *Corollary 9 is an improvement of Theorem 3.2 of Osilike et al. [13] and Theorems 3.1 and 3.2 of Zhang and Xie [17] in the sense that our convergence is *strong* without the requirement that , provided that possesses weakly sequentially continuous duality mappings.

#### Acknowledgments

N. Shahzad gratefully acknowledges research support from the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia.